Internal Resistance of Batteries: Beyond the Textbook for H2 Physics

TL;DR
That "ideal battery" in your circuit diagram? It doesn't exist. Every real battery has internal resistance \(r\) that limits current and wastes power. This guide shows how to measure \(r\) for different battery types, automate data collection with Arduino, and understand why your phone charger gets warm. Essential for scoring in H2 Physics Paper 4's favourite circuit practical.

The Reality Check Your Textbook Skips

Your textbook shows: \(\mathcal{E} = I(R + r)\)

What actually happens:

• Fresh AA battery: \(r ≈ 0.1\) Ω
• After 50% discharge: \(r ≈ 0.3\) Ω
• Cold battery at 0°C: \(r\) doubles
• Old 9V battery: \(r > 10\) Ω

This isn't just exam fodder - it's why your torch dims as batteries die, and why electric cars struggle in winter.

Core Theory (The Exam Version)

For a battery with EMF \(\mathcal{E}\) and internal resistance \(r\):

\[ V = \mathcal{E} - Ir \]

Where:

• \(V\) = Terminal voltage (what you measure)
• \(I\) = Current through circuit
• \(\mathcal{E}\) = EMF (open-circuit voltage)
• \(r\) = Internal resistance

Rearranging: \(V = -rI + \mathcal{E}\)

This is a straight line with:

• Gradient = \(-r\)
• Y-intercept = \(\mathcal{E}\)

Equipment Setup That Actually Works

Basic Circuit

[Battery] --+-- [Ammeter] --+-- [Variable Resistor]
            |               |
            +--[Voltmeter]--+

Smart Equipment Choices

Variable Resistor Options:

1. Resistance box (0-100 Ω): Precise but expensive
2. Rheostat: Smooth variation but needs calibration
3. Multiple fixed resistors: Cheap and reliable
4. Electronic load: Pro-level accuracy

Measurement Tools:

• Digital multimeters (at least 3½ digits)
• Current: 0-2A range typically
• Voltage: 0-20V range
• Arduino + INA219 sensor for automated logging

Testing Different Battery Types

1. AA Alkaline Batteries (1.5V)

Setup specifics:

• Current range: 0-1A (max ~2A short circuit)
• Load resistance: 1-50 Ω
• Expected \(r\): 0.1-0.5 Ω

What you'll discover:

• Fresh Duracell: \(r \approx 0.15\) Ω
• Generic brand: \(r \approx 0.25\) Ω
• Nearly dead: \(r > 1\) Ω

2. 9V Batteries

The surprise: Much higher internal resistance!

• Fresh: \(r \approx 1-2\) Ω
• Half-used: \(r \approx 5-10\) Ω
• Current limit: <500mA practically

Why? Six tiny 1.5V cells in series = 6x the resistance

3. Phone Power Banks

The complex beast:

• Contains lithium cells + protection circuits
• Apparent \(r\) includes electronics
• Output may be regulated (constant V)
• Need USB tester for proper measurements

Typical results:

• Good 10,000mAh bank: \(r_\text{apparent} ≈ 0.2\) Ω
• Includes cable resistance!

Data Collection: Manual vs Automated

Manual Method (Exam-style)

1. Set resistance box to highest value
2. Record \(V\) and \(I\)
3. Decrease \(R\) in ~10 steps
4. Stop before \(I > 1A\) (battery safety)
5. Plot immediately to check linearity

Pro tip: Take readings quickly - batteries heat up!

Arduino Automated Logger

// Simplified code structure
// Uses INA219 current/voltage sensor

#include <INA219.h>

void loop() {
  float voltage = ina.getBusVoltage();
  float current = ina.getCurrent();

  Serial.print(voltage);
  Serial.print(",");
  Serial.println(current);

  delay(100); // 10 readings per second
}

Advantages:

• 100+ data points in seconds
• Captures transient effects
• Export straight to Excel
• Multiple batteries tested quickly

Temperature Effects (The Missing Chapter)

Why Temperature Matters

Internal resistance changes dramatically:

• Room temp (25°C): Baseline \(r\)
• Fridge (4°C): \(r\) increases ~50%
• Freezer (-18°C): \(r\) doubles or more
• Hot car (45°C): \(r\) decreases ~20%

Experimental Setup

1. Seal battery in zip-lock bag
2. Submerge in water bath
3. Wait 15 mins for thermal equilibrium
4. Measure \(r\) at each temperature
5. Plot \(r\) vs \(T\)

Safety: Don't heat above 50°C!

What You'll Find

For alkaline batteries: \[ r(T) ≈ r_{25°C} \times e^{-b(T-25)} \]

Where \(b \approx 0.02\) K⁻¹

Advanced Analysis Techniques

1. Power Transfer Theorem

Maximum power delivered when \(R_\text{load} = r\):

\[ P_\text{max} = \frac{\mathcal{E}^2}{4r} \]

Experiment:

• Vary \(R\) from 0.1r to 10r
• Calculate \(P = I^2R\) for each
• Find peak power point
• Verify \(R_\text{peak} ≈ r\)

2. AC Impedance Method

Internal resistance has components:

• DC resistance (ionic conduction)
• AC impedance (capacitive effects)

Using function generator + oscilloscope:

• Apply 1kHz sine wave
• Measure voltage/current phase
• Calculate complex impedance

3. Transient Response

Connect/disconnect load suddenly:

• Voltage doesn't change instantly
• Time constant \(\tau = rC\) reveals battery capacitance
• Relevant for pulse-discharge applications

Common Mistakes That Cost Marks

1. "My graph curves!"

Causes:

• Battery heating during experiment
• Approaching current limit
• Protection circuit kicking in

Fix: Use lower currents, work quickly

2. "Negative internal resistance?"

You probably:

• Swapped voltage/current axes
• Connected ammeter in parallel
• Have a regulated power bank

3. "Huge error bars"

Check:

• Meter resolution (need 0.01V minimum)
• Contact resistance (clean terminals!)
• Wire resistance (use thick wires)

Data Analysis Template

Excel Setup

Parameters

• A: Resistance (Ω)
• B: Current (A)
• C: Voltage (V)
• D: =B*A \( Check: should ≈ C \)

Graph

• Plot: C (y-axis) vs B (x-axis)
• Trendline: Linear
• Display: Equation + R²

Uncertainty Analysis

For each measurement:

• \(\pu{\delta V = \pm 0.01V}\) (meter resolution)
• \(\pu{\delta I = \pm 0.01A}\) (meter resolution)
• \(\delta r = |\text{gradient}| \times \sqrt{(\frac{\delta V}{V})^2 + (\frac{\delta I}{I})^2}\)

For gradient from graph:

• Use LINEST function for uncertainty
• Or calculate from worst-fit lines

Real-World Applications

Why This Matters Beyond Exams

1. Electric Vehicles
   • Battery pack \(r\) limits acceleration
   • Cold weather range reduction
   • Thermal management critical
2. Solar Systems
   • Battery bank sizing
   • Cable gauge selection
   • Inverter efficiency optimization
3. Portable Electronics
   • Fast-charging limits
   • Battery life estimation
   • Heat generation prediction

Exam-Style Questions You'll Ace

Q1: "Explain why terminal voltage drops under load"

Model answer: Internal resistance causes voltage drop \(Ir\) inside battery. Terminal voltage \(V = \mathcal{E} - Ir\) decreases as current increases.

Q2: "Suggest improvements to reduce uncertainty"

Points to include:

• Use lower currents (reduce heating)
• Automate readings (minimize time)
• Multiple batteries averaged
• 4-wire measurement for low \(r\)

Q3: "Why might \(r\) appear to change during experiment?"

Key factors:

• Temperature rise from \(I^2r\) heating
• Chemical polarization at high currents
• Battery capacity depletion
• Protection circuit activation

Beyond the Basics: Research Extensions

1. Battery Chemistry Comparison

• Alkaline vs NiMH vs Li-ion
• How \(r\) changes with discharge state
• Recovery effects after heavy load

2. Internal Resistance Spectroscopy

• Measure \(r\) vs frequency (1Hz - 10kHz)
• Reveals battery health/age
• Used in battery management systems

3. Four-Wire (Kelvin) Measurement

• Eliminates lead resistance
• Essential for \(r < 0.1\) Ω
• Industry-standard technique

Your Practical Exam Checklist

✓ Check meter zeros before starting
✓ Clean all contacts with alcohol
✓ Start with high \(R\) (low current)
✓ Work quickly to minimize heating
✓ Plot as you go to spot problems
✓ Multiple trials if time allows
✓ Calculate \(r\) from gradient, not individual points
✓ Discuss temperature in evaluation

Master this practical and you'll not only secure full marks - you'll understand why your phone battery meter lies, why jump-starting works, and why battery technology remains the bottleneck in our electric future.